Tag Archives: Knight’s Tour

Continuous Line Artist view of Haken’s Gordian Knot in “Unknot Hall of Fame”.

Peter Prevos, has included me in his “Unknot Hall of Fame”, within an article on his website about “The Art and Magic of the Trivial Knot”, which also explains many technical aspects of the trivial knot and how magicians have incorporated those ideas. There are designs, in the Hall of Fame, by Goeritz, Thistlewaite, Ochiai and Haken as well as art by Vanuatu and myself and reference to James Sienna. Have a look on – https://horizonofreason.com/science/unknot-gallery/

I had already done my painting when I saw a post on Mathoverflow website “Are there any very hard unknots?” by mathematician Timothy Gowers – https://mathoverflow.net/questions/53471/are-there-any=very-hard-unknots

I responded with posts on my website on in June 2015 and a later update in May 2018.

Noboru Ito, mathematician now at the of University of Tokyo, contacted me in February 2016 about his near completed book “Knot Projections” and my article is referenced in the Preface. “It was very helpful”.

Tomasz Mrowka, mathematician at the Massachusetts Institute of Technology, asked in November 2017 for a copy of my painting, as “it’s really quite striking and would love to hang it in my office”.

David Eppstein, computer scientist and mathematician at university of California, Irvine, featured my painting on his website in November 2018 “Mick Burton, an artist known for drawings that use a single continuous line to create the impression of complex and naturalistic shapes, looks at knot theory, self-overlapping curves, and the visualization of Seifert surfaces.” I had to look in Noboru Ito’s book to check out Seifert surfaces !

In essence my painting of Haken’s Gordian Knot is another example of me finding a well known structure which I can apply my continuous line knowledge and experience to. The way that nature can work in these structures often surprises me.

This is separate to my ongoing art work of producing single continuous lines and colouring based upon interesting subjects – which can be animals, landscapes, portraits, still life and abstracts.

Other examples of looking at structures have been –

Four colour theory maps, where my overdraw method could divide a map up into two streams of alternate colours, hence the four colours. I corresponded with mathematicians Robin Wilson and Fred Holroyd in the mid 1970’s. See my post in August 2015 on Skydiver patterns and my Four Colour Theory.

The artist Escher’s favourite tile at the Alhambra in Spain, which I realised had two continuous lines running through it. I saw that the artist could have made in into a single line with two small alterations. See my posts of April 2015.

Knights moves on a chess board starting and finishing at the same square and landing once on all the other squares. See my 1974 picture in the Gallery 1965-1974.

I am always on the look out for new structures which are suitable.

“Vortex” by David Kilpatrick. Single Continuous Line and Alternate Overdraw colouring.

Vortex, David Kilpatrick. flat,1000x1000,075,f.u1

“Vortex” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line using the Alternate Overdraw method to allocate colours.   March 2017.   Mick Burton blog.

I have been exchanging ideas with David Kilpatrick recently and he has agreed to let me put some of his pictures in my blog.  “Vortex” stands out to me, as I have been a fan of Vorticism for many years.  He has used Alternate Overdraw to allocate colours in sequence and it has worked well.

David’s design gives the impression of a sheet of plastic, coloured green on one side and red on the other, and each twist showing the other side.  With overlaps you get darker greens or darker reds.  Four internal areas let the background shine through.  The whole thing is very natural, including David’s own style of patchy colour radiating outwards.

Next is David’s “Knight’s Tour” which he is still working on.

David Kilpatrick knights tour. image011

“Knights Tour” by David Kilpatrick, artist from Atherton, Australia.   Single Continuous Line based upon moves of a knight and using Alternate Overdraw to allocate colour sequence.   April 2017.  Mick Burton blog.

I did a Single Continuous Line “Knight’s Moves on a Chessboard” in 1973 (see Gallery 1965-74) with the intention of colouring it, but never tackled it properly.  One of the problems was the number of fiddly small areas.  It led to my “Knight’s Tour Fragments” instead (see my previous post on 16.2.2017).

But now we have GIMP!  David said that he used this to move the lines about on his “Knight’s Tour”.  I googled GIMP and it means “GNU Image Manipulation Program”.  Some areas are still fairly small but he has produced a vibrant structure.

David says that these are trial colours (I presume from GIMP) and he intends to work out an improved scale of colours in his own style.

However, the colours shown already demonstrate the natural balance inherent in the Alternate Overdraw colour allocation.  The composition suggests to me an island with yellow “beaches” as well as reds within opposite “volcanic” zones.

There is a choice regarding background, which would naturally be the same colour as the light blue internal areas and result in a surrounding “sea”, or it could be left white as shown above.

I look forward to seeing the final version, which I am sure will be another splendid example of Vorticism.

Another picture that caught my eye was his “The Pram” which is based on a magic rectangle.

“The Pram” by artist David Kilpatrick, from Atherton, Australia.   Based on a Magic Rectangle. 2015.    Mick Burton blog.

This pram picture has lots of line ends in it and makes me want to attempt one myself using a Continuous Line animal.  Such a design would make you want to connect up so many loose ends.  My Spherical pictures already do this to an extent, as I take a line out of the picture at one side and bring it back in at the corresponding opposite side.

I think that David chose the positions of the displaced squares in a sort of random way.  Maybe I would want to be confident that I could move them around, in the way you could on the movable squares game of my childhood, and get back to the actual original picture.

You can see much more of the art of David Kilpatrick on

https://www.redbubble.com/people/fnqkid